PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Proof.
Consider 0 ≤ Var(X 1 − tX 2 ) (for any constant t)
= σ 2 1 + t 2 σ 2 2 + 2tσ 12 (by Corollary 1).
Hence quadratic,
q(t) = σ 2 2t 2 + 2σ 12 t + σ 2 1 ≥ 0, for all t.
Hence q(t) has either no real roots or
one real root.
* no real roots if ∆ = b 2 − 4ac < 0
* one real root if ∆ = b 2 − 4ac = 0.
Hence ∆ ≤ 0
⇒ 4σ 2 12 − 4σ 2 2σ 2 1 ≤ 0
⇒ 4σ 2 12 ≤ 4σ 2 1σ 2 2
(σ’s +ve)
⇒ σ2 12
σ 2 1σ 2 2
≤ 1
⇒ ρ 2 ≤ 1
⇒ |ρ| ≤ 1.
Suppose |ρ| = 1 ⇒ ∆ = 0
⇒ there is single t 0 such that:
0 = q(t 0 ) = Var(X 1 − t 0 X 2 ),
i.e., X 1 = t 0 X 2 + c with probability 1.
Theorem. 1.9.4
If X 1 , X 2 are independent, and Var(X 1 ), Var(X 2 ) exist, then Cov(X 1 , X 2 ) = 0.
Proof. (Continuous)
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